If you want to find "a possible range of values where this index could be useful" then you should examine the continuous relation of your index to outcome. Looking for a single cut point will not accomplish that task. See this question for discussion in the general regression context and this question in a broader machine-learning context.

For a novel index it's particularly important to understand how its values over its range are related to outcome. Maybe you will find that there is some upper or lower limit beyond which changes in the index don't matter, but you won't learn that unless you look in detail. It's also important to see whether the index adds anything to already established prognostic variables, something you can't do with a single-predictor Cox model.

Furthermore, the word "optimal" can hide a lot of assumptions. In the classification context, it can make the assumption that false positives have the same costs as false negatives, which isn't always the case. See this answer for discussion of cut points in the context of Cox models.

If you nevertheless are compelled to look for a cut point, your proposed method seems to be essentially that used by the cutp function in the R survMisc package. I would recommend that you try your cut point selection on multiple bootstrap samples from your data, as that best mimics repeated sampling from the underlying population. Unless your numeric index is hiding some true dichotomy in an underlying phenomenon, my hunch is that you will find a pretty wide range of "optimal" cut points, however defined, among those repeated samples. The bootstrap results at least will show your readers how much reliability to associate with the cut point value that you propose.

If you want to find "a possible range of values where this index could be useful" then you should examine the continuous relation of your index to outcome. Looking for a single cut point will not accomplish that task. See this question for discussion in the general regression context and this question in a broader machine-learning context.

For a novel index it's particularly important to understand how its values over its range are related to outcome. Maybe you will find that there is some upper or lower limit beyond which changes in the index don't matter, but you won't learn that unless you look in detail. It's also important to see whether the index adds anything to already established prognostic variables, something you can't do with a single-predictor Cox model.

Furthermore, the word "optimal" can hide a lot of assumptions. In the classification context, it can make the assumption that false positives have the same costs as false negatives, which isn't always the case. See this answer for discussion of cut points in the context of Cox models.

If you nevertheless are compelled to look for a cut point, your proposed method seems to be essentially that used by the cutp function in the R survMisc package. I would recommend that you try your cut point selection on multiple bootstrap samples from your data, as that best mimics repeated sampling from the underlying population. Unless your numeric index is hiding some true dichotomy in an underlying phenomenon, my hunch is that you will find a pretty wide range of "optimal" cut points, however defined, among those repeated samples. The bootstrap results at least will show your readers how much reliability to associate with the cut point value that you propose.